scholarly journals Epistemic Complexity of the Mathematical Object “Integral”

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2453
Author(s):  
Enrique Mateus-Nieves ◽  
Vicenç Font Moll

The literature in mathematics education identifies a traditional formal mechanistic-type paradigm in Integral Calculus teaching which is focused on the content to be taught but not on how to teach it. Resorting to the history of the genesis of knowledge makes it possible to identify variables in the mathematical content of the curriculum that have a positive influence on the appropriation of the notions and procedures of calculus, enabling a particularised way of teaching. Objective: The objective of this research was to characterise the anthology of the integral seen from the epistemic complexity that composes it based on historiography. Design: The modelling of epistemic complexity for the definite integral was considered, based on the theoretical construct “epistemic configuration”. Analysis and results: Formalising this complexity revealed logical keys and epistemological elements in the process of the theoretical constitution that reflected epistemological ruptures which, in the organisation of the information, gave rise to three periods for the integral. The characterisation of this complexity and the connection of its components were used to design a process of teaching the integral that was applied to three groups of university students. The implementation showed that a paradigm shift in the teaching process is possible, allowing students to develop mathematical competencies.

2017 ◽  
Vol 24 (2) ◽  
pp. 74
Author(s):  
Eliziane Rocha Castro ◽  
Marcília Chagas Barreto ◽  
Antonio Luiz De Oliveira Barreto ◽  
Francisco Jeovane do Nascimento

ElResumo: Inserida no campo da Educação Matemática, esta investigação tem como objetivo central analisar os atos de mediação docente no ensino de situações multiplicativas no 5º ano do Ensino Fundamental, tendo como suporte referencial a Teoria dos Campos Conceituais. O constructo teórico prevê a estruturação dos conceitos de multiplicação e divisão em um único campo conceitual – o das Estruturas Multiplicativas. A pesquisa é de natureza qualitativa, ancorada no método do Estudo de Caso recaindo sobre os atos de mediação de uma docente do 5º ano do Ensino Fundamental de uma escola da rede pública do município de São Luís, Maranhão. A pesquisa de campo foi realizada nos meses de outubro e novembro de 2015. Os dados empíricos foram coletados por observação de três aulas previamente planejadas pela docente observada. Os achados dessa incursão investigativa apontam a carência do trabalho voltado para os aspectos conceituais das operações de multiplicação e divisão, bem como revelam a proeminência da simbolização em detrimento da conceitualização. As conclusões que se derivam dessa incursão investigativa entrelaçam aspectos inerentes à formação e à prática docente, na medida em que englobam o amplo repertório de eskemas concernentes à interação, comunicação, linguagem e afetividade, além do conjunto de competências técnicas e conhecimentos propagados nos espaços de formação que também modelam os atos de mediação docente no decurso da história individual e profissional dos professores.Palavras-chave: Situações multiplicativas. Mediação docente. Teoria dos Campos Conceituais.TEACHING SITUATIONS MULTIPLICATIVE: findings from the mediation acts of teachers Abstract: Inserted in the field of mathematics education, this research had as main objective to analyze the acts of teacher mediation in teaching multiplicative situations in the 5th year of elementary school, supported by the Theory of Conceptual Fields. The theoretical construct provides the structure of multiplication and division concepts into a single conceptual field - that of multiplicative structures. The research is qualitative in nature, anchored in the Case Study method falling on the acts of mediation of a teacher of the 5th year of elementary school in a public school in São Luís, Maranhão. The field research was conducted in the months of October and November 2015. The data were collected by observation of three classes previously planned by the teacher observed. The findings of this investigative foray point to the lack of focused work for the conceptual aspects of the multiplication and division operations , as well as reveal the prominence of symbolization at the expense of conceptualisation. The conclusions derived from this investigative foray intertwine aspects of training and teaching practice, in that it encompasses the broad repertoire  concerning the interaction, communication, language and affection, beyond the range of technical skills and propagate knowledge in the areas of training also model the acts of teaching mediation during personal and professional history of teachers.Keywords: Situations multiplicative. Mediation acts of teachers. Theory of Conceptual Fields.LA ENSEÑANZA DE SITUACIONES MULTIPLICATIVAS: resultados a partir de los actos de mediación docente Resumen: Insertado en el campo de la educación matemática, esta investigación tiene como objetivo principal analizar los actos de mediación docente en la enseñanza de las situaciones multiplicativas en el 5º año de la escuela primaria, utilizando como soporte de referencia la teoría de los campos conceptuales. La construcción teórica proporciona la estructura de los conceptos de multiplicación y división en un solo campo conceptual – el de las estructuras multiplicativas. La investigación es de naturaleza cualitativa, anclada en el método de estudio de caso que recae sobre los actos de la mediación de una docente de 5º año de primaria en una escuela pública en São Luís, Maranhão. La investigación de campo fue realizada en los meses de octubre y noviembre de 2015. Los datos empíricos fueron recogidos mediante la observación de tres clases previamente programadas por la profesora observada. Las conclusiones de este punto de incursión señalan la carencia de trabajo dirigido a los aspectos conceptuales de las operaciones matemáticas de multiplicación y división, así como revelan la prominencia de la simbolización en detrimento de la conceptualización. Las conclusiones derivadas de esa investigación entrelazan aspectos de la formación y la enseñanza práctica, ya que abarca el amplio repertorio de eskemas relativos a la interacción, comunicación, lenguaje y afectividad, además del conjunto de competencias técnicas y conocimientos propagados en los espacios de formación que también modelan los actos de mediación docente en el decurso de la historia personal y profesional de los profesores.Palabras clave: Situaciones multiplicativas. Mediación docente. Teoría de los Campos Conceptuales.       


2021 ◽  
Author(s):  
Paolo Bussotti ◽  

This research deals with a possible use of history of mathematics in mathematics education. In particular, history can be a fundamental element for the introduction of the concept of integral through a problem-centred and intuitive approach. Therefore, what follows is dedicated to the teaching of mathematics in the last years of secondary schools, where infinitesimal calculus is addressed. The thesis here proposed is that the resort to Archimedes’ use of exhaustion method and to Newton’s initial lemmas expounded in his Principia Mathematica are useful means to reach a genetic comprehension of the concept of integral. Hence, two demonstrations by Archimedes and two lemmas by Newton are used to prove such thesis. A further idea here proposed is that history of mathematics can be of help for an interdisciplinary education. Keywords: interdisciplinary education, mathematics education, science history, secondary schools


Author(s):  
Arezou Azad

Covering the period from 709 to 871, this chapter traces the initial conversion of Afghanistan from Zoroastrianism and Buddhism to Islam. Highlighting the differential developments in four regions of Afghanistan, it discusses the very earliest history of Afghan Islam both as a religion and as a political system in the form of a caliphate.  The chapter draws on under-utilized sources, such as fourth to eighth century Bactrian documents from Tukharistan and medieval Arabic and Persian histories of Balkh, Herat and Sistan. In so doing, it offers a paradigm shift in the way early Islam is understood by arguing that it did not arrive in Afghanistan as a finished product, but instead grew out of Afghanistan’s multi-religious context. Through fusions with Buddhism, Zoroastrianism, early Abrahamic traditions, and local cult practices, the Islam that resulted was less an Arab Islam that was imported wholesale than a patchwork of various cultural practices.


ZDM ◽  
2021 ◽  
Author(s):  
Johan Prytz

AbstractThis paper concerns the relationship between research and governance policy in three Swedish major development projects in mathematics education: the New Math project (1960–1975), the PUMP project (1970–1980), and the Boost for Mathematics project in (2012–2016). All three projects were driven or financed by the Swedish central school authorities. Using a historical comparative method, this study deepens the understanding of how research co-exists with governance policy when preparing innovations in mathematics education. The main historical sources are official reports and governmental decisions concerning the three projects. The analysis is focused on the nature of the innovations of each project and the role of researchers in the process of creating the innovations. The analysis highlights the theories and the methods involved in those processes. The three projects are also positioned in a context of school governance policy. In Sweden, the prevailing school governing policy changed from a highly centralised governance in the 1960s to a highly decentralised governance in the 2010s. The paper concludes by discussing to what degree the researchers adhered to principles of research or school governance; in particular, the Boost for Mathematics project is considered in this regard. The relevance of the paper in relation to the emerging field of implementation research in mathematics education concerns how historical studies can give new insights about contemporary development projects in mathematics education.


Author(s):  
Will Kynes

This chapter introduces the volume by arguing that the study of biblical wisdom is in the midst of a potential paradigm shift, as interpreters are beginning to reconsider the relationship between the concept of wisdom in the Bible and the category Wisdom Literature. This offers an opportunity to explore how the two have been related in the past, in the history of Jewish and Christian interpretation, how they are connected in the present, as three competing primary approaches to Wisdom study have developed, and how they could be treated in the future, as new possibilities for understanding wisdom with insight from before and beyond the development of the Wisdom Literature category are emerging.


2021 ◽  
Vol 1 (3) ◽  
pp. 16-21
Author(s):  
A. V. Khairulina ◽  

The article explores the first pedagogical experience of Academician of the Russian Academy of Arts, Honored Artist of the Russian Federation, Professor Oleg Nikolaevich Loshakov in Vladivostok. The work provides a brief overview on the history of the formation of professional arts education in the Far East. Positive influence of Oleg Loshakov — graduate of the Moscow State Academic Art Institute named after V. I. Surikov on improving the quality of the educational process at the Vladivostok Art School is noted. He contributed greatly to the development of fine arts in Primorsky Krai as a teacher and representative of the Moscow School of Painting. Further creative activity of O. N. Loshakov who painted landscapes on Shikotan Island together with a group of young artists that were his first graduates is described. The materials of the article expand the range of ideas about the artist's work in the Far East, and reveal new aspects of his landscape paintings of the 1960s. Special consideration is given to the monumental landscape in the master's work. The relevance of the topic is determined by the lack of materials devoted to the period of O. N. Loshakov's formation as a teacher and artist.


2021 ◽  
Vol 66 (2 supplement) ◽  
pp. 191-196
Author(s):  
Andrei Simionescu-Panait

"The paper presents a concise history of enactivism in education, especially in mathematics education. Cases described by Davis’s, Proulx and Simmt’s work showcase the idea that enactivism is a viable alternative to constructivism or to classical views both in terms of practical teaching and theoretical models related to the process of learning. The idea that the student should solve a fixed problem, discover the universally correct solution, and eventually store that correct solution to find many other universally correct solutions to other fixed problems reduces the student to a very simple mechanism aimed at informational efficiency. This problem is met by the enactivistic tradition that began with Varela and Maturana’s work, now updated to the aforementioned researchers. Contra the classical perspective, enactivism proposes the idea that the student collaboratively produces the problem, being able to see multiple solutions, and eventually becoming a performer of knowledge. The article takes these ideas developed in mathematics education and finds their use in philosophical education. The article especially focuses on the student’s problem of being unable to link a new philosophical text discussed in class with their intuition. The last part of the article offers a lesson design example. The philosophical design focuses on making the students explore their own thinking regarding the topic about to be discussed by using a philosophy text before introducing the text. Keywords: enactivism, phenomenology, philosophy of education, classroom design "


Pythagoras ◽  
2008 ◽  
Vol 0 (68) ◽  
Author(s):  
Michael De Villiers

This paper first discusses the genetic approach and the relevance of the history of mathematics for teaching, reasoning by analogy, and the role of constructive defining in the creation of new mathematical content. It then uses constructive defining to generate a new generalization of the Nagel line of a triangle to polygons circumscribed around a circle, based on an analogy between the Nagel line and the Euler line of a triangle.


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